Growth rate modeling and identification in the crystallization of polymers

Nucleation and growth mechanisms are important kinetics of the phase transformation model which arises in the crystallization of polymer materials. In each stage, the nucleation rate and growth rate are crucial coefficients describing the kinetics of the process as well as the properties of the specimens. Moreover, the identification of these physical parameters describing the nucleation or the growth mechanisms is essential for controlling the crystallization of polymers and so is a significant subject also from an industrial viewpoint. In this paper, we show that we can re-formulate the time cone approach of Cahn (1996 Mater. Res. Soc. Symp. Proc. 398 425–37) by a hyperbolic governing equation with the heterogeneous nucleation rate and spatially homogeneous growth rate. Then, on the basis of the hyperbolic equation, we investigate an inverse problem of determining the growth rate for an isothermal one-dimensional specimen. Our inverse problem is an inverse coefficient problem for a hyperbolic equation which is highly nonlinear with respect to the observation data. A two-step Tikhonov-type regularization method is proposed to reconstruct the growth rate provided with the final noisy observation data. Numerical prototype examples are presented to illustrate the validity and effectiveness of the proposed scheme. (paper)